Rhapsody in R

Exploring the intersection of probability and music

John Zito

Duke Stats

2025-08-09

About me

• It’s my sixth go-around teaching intro probability;
• I’m a tad weary of coins, dice, playing cards, etc.

Trying to mix it up

Plenty of applications from the natural and social sciences:

• Actuarial mathematics;
• Contested elections;
• Dating apps;
• DNA sequencing;
• Expert witness testimony;
• Extreme weather;

• I Ching divination;
• Investment risk and return;
• Language models;
• Papal conclaves;
• Prediction markets;
• Quantum mechanics.

But what about the arts?

Stochastic music

AKA: chance or aleatoric music.

• leaving some aspect of the composition up in the air until the moment of performance;
• ⭐ simulating a random process to determine what notes to write down.

Iannis Xenakis (1922 - 2001)

• Studied both traditional western music and CS, statistical mechanics, stochastic processes, etc;
• Incorporated these ideas into his compositional process.

Pithoprakta (1956)

Think of each member of a 46-piece string orchestra as a Brownian particle drifting up and down the staff:

“[E]very member of the orchestra is on the honor system.” Leonard Bernstein

Pithoprakta (1956)

Well …it’s the thought that counts.

Main idea

Thought: Can I prompt students to use what they know about probability distributions and simulation to write their own pieces of stochastic music?

Worry: Depends. Can you work with music in R?

Renfei Mao’s gm package

“grammar of music”

• Represent music in R with a ggplot2-style interface;

“generate music”

• integration with MuseScore generates sheet music and MIDI playback.

Ravel: Prélude in A Minor, M. 65

Let’s transcribe it!

The starting point is always the same

We will add layers to this:

library(gm)

Attaching package: 'gm'

The following object is masked from 'package:methods':

    show

prelude <- Music()

Analogous to this:

library(ggplot2)

myplot <- ggplot()

Add meter and tempo

Add meter and tempo

prelude <- Music() + 
  Meter(3, 4) + 
  Tempo(60)

Add the right hand part

Brief aside: scientific pitch notation

"F#5" means the F♯ in the fifth octave on the piano, etc.

Add the right hand part

right_hand <- Line(
  pitches = list(NA, "E5", "F#5", "D5", "E5", "F#5",
                 "B5", "G5", "D5", 
                 "E5", "G4",
                 c("D4", "F#4", "B4"), c("C#4", "F4", "C5"),
                 c("G3", "C4", "E4"), c("D#4", "G4"), c("C4", "E4"))
)

prelude <- Music() + 
  Meter(3, 4) + 
  Tempo(60) +
  right_hand

Add the right hand part

gm::show(prelude)

(BTW: this plays nice with Quarto off-the-shelf.)

Adjust the note values

right_hand <- Line(
  pitches = list(NA, "E5", "F#5", "D5", "E5", "F#5",
                 "B5", "G5", "D5", 
                 "E5", "G4",
                 c("D4", "F#4", "B4"), c("C#4", "F4", "C5"),
                 c("G3", "C4", "E4"), c("D#4", "G4"), c("C4", "E4")),
  durations = c(0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 
                1, 1, 2, 
                1, 1, 
                2, 1, 
                1.5, 0.5, 1)
)

prelude <- Music() + 
  Meter(3, 4) + 
  Tempo(60) +
  right_hand

Adjust the note values

gm::show(prelude)

Add the left hand part

Add the left hand part

left_hand <- Line(
  pitches = c("A2", "E3", "G3", "C4", "E4", 
              "G4", "C5", NA, 
              NA, "E2", "E3", "F#3", "G#3",
              NA, "A2", "E3", "B3", "G3"),
  durations = c(rep(0.5, 7), 2, 
                rep(0.5, 4), 1, 
                rep(0.5, 4), 1),
  bar = 2, offset = 0.5
)

prelude <- Music() + 
  Meter(3, 4) + 
  Tempo(60) +
  right_hand + 
  left_hand + 
  Clef("F")

Note: rests correspond to missing values (NA) in the line.

Add the left hand part

gm::show(prelude)

Add dynamics

Add dynamics

prelude <- Music() + 
  Meter(3, 4) + 
  Tempo(60) +
  right_hand + 
  Dynamic("p", 1) + 
  left_hand + 
  Clef("F") +
  Dynamic("p", 1)

Add expressive indications

Add expressive indications

prelude <- Music() + 
  Meter(3, 4) + 
  Tempo(60) +
  right_hand + 
  Dynamic("p", 1) + 
  Slur(2, 11) + 
  Slur(12, 16) + 
  left_hand + 
  Clef("F") +
  Dynamic("p", 1) + 
  Slur(1, 7) + 
  Slur(10, 13) + 
  Slur(15, 18) + 
  Pedal(1, 7)

Pretty close!

gm::show(prelude)

Change the instrumentation (optional)

prelude <- Music() + 
  Meter(3, 4) + 
  Tempo(65) + 
  right_hand + 
  Dynamic("p", 1) + 
  Slur(2, 11) + 
  Slur(12, 16) + 
  Instrument(47) + # Harp!
  left_hand + 
  Clef("F") +
  Dynamic("p", 1) + 
  Slur(1, 7) + 
  Slur(10, 13) + 
  Slur(15, 18) + 
  Instrument(43) # Cello!

Change the instrumentation (optional)

?Instrument

1. Acoustic Grand Piano
2. Bright Acoustic Piano
3. Electric Grand Piano
4. Honky-Tonk Piano
5. Electric Piano 1
6. Electric Piano 2
7. Harpsichord
8. Clavinet

9. Celesta
10. Glockenspiel
11. Music Box
12. Vibraphone
13. Marimba
14. Xylophone
15. Tubular Bells

… and so on

Change the instrumentation (optional)

gm::show(prelude)

Summary: the gm package

A ggplot2-style interface for music:

prelude <- Music() + 
  Meter(3, 4) + 
  Tempo(65) + 
  right_hand + 
  Dynamic("p", 1) + 
  Slur(2, 11) + 
  Slur(12, 16) + 
  Instrument(47) +
  left_hand + 
  Clef("F") +
  Dynamic("p", 1) + 
  Slur(1, 7) + 
  Slur(10, 13) + 
  Slur(15, 18) + 
  Instrument(43) 

gm::show(prelude)

Thanks Renfei!

Now, let’s write some crazy music!

Every pitch on the piano

pitches <- c("C", "C#", "D", "D#", "E", "F", "F#", "G", "G#", "A", "A#", "B")
octaves <- 1:7
all_pitches <- c("A0", "A#0", "B0", 
                 paste(rep(pitches, length(octaves)), 
                       sort(rep(octaves, length(pitches))), 
                       sep = ""), 
                 "C8")
all_pitches

 [1] "A0"  "A#0" "B0"  "C1"  "C#1" "D1"  "D#1" "E1"  "F1"  "F#1" "G1"  "G#1"
[13] "A1"  "A#1" "B1"  "C2"  "C#2" "D2"  "D#2" "E2"  "F2"  "F#2" "G2"  "G#2"
[25] "A2"  "A#2" "B2"  "C3"  "C#3" "D3"  "D#3" "E3"  "F3"  "F#3" "G3"  "G#3"
[37] "A3"  "A#3" "B3"  "C4"  "C#4" "D4"  "D#4" "E4"  "F4"  "F#4" "G4"  "G#4"
[49] "A4"  "A#4" "B4"  "C5"  "C#5" "D5"  "D#5" "E5"  "F5"  "F#5" "G5"  "G#5"
[61] "A5"  "A#5" "B5"  "C6"  "C#6" "D6"  "D#6" "E6"  "F6"  "F#6" "G6"  "G#6"
[73] "A6"  "A#6" "B6"  "C7"  "C#7" "D7"  "D#7" "E7"  "F7"  "F#7" "G7"  "G#7"
[85] "A7"  "A#7" "B7"  "C8" 

The simplest piece of stochastic music

Sample the pitches with replacement:

set.seed(8675309)

line1 <- Line(
  pitches = sample(all_pitches, 64, replace = TRUE),
  durations = .25
)

line2 <- Line(
  pitches = sample(all_pitches, 64, replace = TRUE),
  durations = .25
)

kitten <- Music() + 
  Meter(4, 4) + 
  Tempo(120) + 
  line1 + 
  line2 + 
  Dynamic("p", 1) +
  Dynamic("ffff", 64) +
  Hairpin("<", 2, 63)

(Angry) Kitten on the keys

What is a melody? (wrong answers only)

It’s a time series!

Game:

• Fix a key (C minor) and a stable accompaniment (waltz);
• Simulate a melody from a Markov chain.

Un(Markov)chained melodies

From Nierhaus’ Algorithmic Composition (2009 Springer):

Un(Markov)chained melodies

A first-order Markov chain on the C minor scale is described by its transition probabilities: given the note we’re on right now, what are the probabilities for the next note to come?

Loads of options to explore:

• uniform (sampling with replacement, again);
• upweight leaps into “chord tones;”
• upweight leaps between chord tones (arpeggio-like);
• upweight leaps into the tonic (“resolutions”);
• random walk (flip a coin every beat and move \(\pm1\)).

IID Melody

Random walk melody

Favors leaps between chord tones

The Jacobson method

Take an existing piece of music and “add noise.”

So this:

\[ \mathbf{y}=f(\mathbf{x})+\boldsymbol{\varepsilon}, \]

only…it’s music?

Name that tune

The students’ mission

• Write your own piece of stochastic music;
• Explore the relationship between the probability rules governing the system and the sound of the resulting music;
• Key words:
  • Play;
  • Surprise yourself.

Randomizing the rhythm

I just like how this looks

Playing with texture

(John, please don’t forget to lower the volume on this.)

They called it “Nightmare Ballet”

Just trolling me

Warning in positions[i] <- new_pos: number of items to replace is not a
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Not quite random, and yet…

Future improvements

• Can gm handle any Hz, and not just the discrete pitches of the Western system?
• How closely can I replicate Xenakis’ methods?
• Play with Markov melodies based on estimated transition probabilities from famous composers;
• Randomize other features: harmony, rhythm, articulation, orchestration, dynamics, etc;
• Students need more time and better coaching/inspiration;
• How can I make it sillier?

Quick takeaways

• Intro probability is a mixed audience. I aim for maximum variety in the examples and applications;
• If you as an instructor have any creative or artistic interests, mix them in. You will have a tremendous amount of fun;
• Activities like this totally tilt the material on its ear;
• The gm package is so much fun to play with!

Thank you!